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x-intercepts of a Quadratic Function

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x-intercepts of a Quadratic Function

Study Guide

Key Definition

The x-intercepts of a quadratic function $f(x) = ax^2 + bx + c$ are the solutions to the equation $ax^2 + bx + c = 0$.

Important Notes

The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives the solutions for x-intercepts.
If $b^2 - 4ac > 0$, the function has two real and distinct x-intercepts.
If $b^2 - 4ac = 0$, the function has one real x-intercept.
If $b^2 - 4ac < 0$, the function has no real x-intercepts.
The expression $b^2 - 4ac$ is called the discriminant.

Mathematical Notation

$-$ represents subtraction

$\pm$ represents plus or minus

$\sqrt{a^2 + b^2}$ represents the square root of a^2 + b^2

$\frac{p}{q}$ represents the fraction p over q

Remember to use proper notation when solving problems

Why It Works

The quadratic formula derives from the process of completing the square on the equation $ax^2 + bx + c = 0$.

Remember

The discriminant $b^2 - 4ac$ helps determine the number and type of roots.

Quick Reference

Quadratic Formula:$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Understanding x-intercepts of a Quadratic Function

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To find the x-intercepts of $f(x)=ax^2+bx+c$, set the function equal to zero and factor the quadratic if possible. For example, $x^2-5x+6=(x-2)(x-3)=0$, so x = 2 or x = 3.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What are the x-intercepts of the quadratic function $f(x) = x^2 - x - 6$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A ball is thrown upwards, and its height is modeled by $h(t) = -16t^2 + 32t + 48$. When does it hit the ground?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the x-intercepts if the discriminant is less than zero in $ax^2 + bx + c = 0$?

Challenge Quiz

Single Choice Quiz

Advanced

Find the x-intercepts of $f(x) = 2x^2 - 4x + 1$.

Recap

Watch & Learn

Review key concepts and takeaways